Graph each function. Give the domain and range.

Question

ƒ (x) = - (\frac{1}{3})^{(x + 3)} - 2

Accepted Answer

Welcome back everyone. In this problem, we want to draw the graph of F FX equals negative or quarters to the power of X plus one minus four. If we look at our answer choices, A has one possible graph where the domain is between negative and positive infinity and ranges from negative infinity to negative four. B has another graph that says the domain is from negative to positive infinity and ranges from negative infinity to negative four C has another graph where it says the domain is from negative infinity to negative four and the ranges from negative to positive infinity. And D has a graph that says the domain is from negative infinity to negative four and it ranges from negative to positive infinity. Now, if we're going to figure out which one of these graphs are correct, let's try to do a sketch of our graph and then compare it to our answer choices. OK. Now notice here, OK that in this case, if we're going to do a graph of F FX, it would probably be best to break down our basic function and build on it. Now, if we look at it here at the heart of our function. Really, we have a horizontal Asymptote. OK? And no, if we take that function. In other words, if we let F of X be equal to a quarter raised to the power of X, OK, we can plot this graph and then build from there based on the additional information we have for F of X. OK. So let this be the basic function of F of X. Now first, it would be good to find Y intercept for our Y intercept. We know that X equals zero. So the Y intercept is going to be equal to a quarter raised to the power of zero, which is equal to one, thus 01 is our Y intercept. Now, if we take this a step further, we can plot a few points for F FX for the basic function of F FX. That is OK. So if we do a table here of X and Y values, and let's say they are between negative two and positive two, if we substitute our X values into our function in the same way, we did find our Y intercept, then when X is negative two, Y is 16, when it's negative one, Y is four, when it's zero, it's one, when it's one, it's a quarter and when it's two, it's 1/16. So now we have an idea of what an Asymptote should look like. So in this sense, we can do a quick sketch. Now, if we do a sketch here, OK, do a quick sketch here, then that means our GUS is going to probably look something like this. And I remember and as ASYM to never touches the axis. In this case, a quarter is the power of X will never touch the X axis, but it will get really close to it. I'm just trying to do a smooth sketch here. It's not exactly smooth, but in essence, we know what our basic function should look like. So now let's try to start building on our function if we look at the power one is added to X. So that means we're going to shift the graph of the basic function horizontally one unit to the left. So let's say we would move it here. OK? And no, that means our Y intercept OK would actually be at negative 11. Next we notice that we make our term with 1/4 OK? A negative value. OK. So that means essentially we're reflecting it in the X axis. And now when we do that, um that means our curve or as to it is probably gonna look something like this. OK? Where now instead of negative 11, the part that was initially the way I intercept, OK. This point that we already know is not going to be negative one, negative one. And we're just keeping note of that point to help us know where we are in terms of position. Now, finally, in our function, we know we subtract four. So since four is subtracted from the function that means we shift the graph obtained in our previous step downward by four units. So we're gonna take this graph, we're gonna bring it down by four units means it's gonna be probably somewhere here. And now this point OK, this point that we originally had that was negative 11 is not going to be negative one negative five. OK. So that's where our point is going to be. And now this is going to be a sketch of a PX equals negative of four. OK. The negative of fourth to the power of X plus one minus four. Now if we think about it, OK, then we know that the domain of the function is going to be all possible real value. So that's from negative infinity to positive infinity. OK. So this would be the domain for our function here. But when it comes to our range, OK, we know that in the same way the horizontal Asymptote would never go below the X axis. In this case, this Asymptote for a quarter, negative a quarter of uh sorry, negative a quarter is to the power of X plus one minus four will never go above negative four. OK? And it will never be negative four, it will never touch negative four. So that means then that the domain is going to be from negative to positive infinity. And the range OK is going to go from negative infinity to negative four. So now the question is which of these graphs would best represent this sketch? Well, when we compare, we should find that A OK. A is the correct answer. Thanks for watching everyone. I hope this video helped.

Graph each function. Give the domain and range. $ƒ (x) = - (\frac{1}{3})^{(x + 3)} - 2$

Key Concepts

Exponential Functions

Domain and Range

Transformations of Functions

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